Creating Camera Sway
The best and easy way I found to make easy camera sways is by using the graph editor.

Steps:


Press I with the camera selected and add a rotation keyframe.
Open up the graph editor and select Camera Y Euler Rotation for example.
Click the N button to pull up the side bar
Click Add Modifier and select Noise
Now you have very fast camera shake, but you want swaying, so set the Scale to something like 10-20. And set the Strength to something like 0.2 - 0.05
Now you could apply this to the other rotations or even some location, but remember to randomize each noise by changing the Phase value for each noise modifier.


Hope that helps!